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arXiv:quant-ph/0506012 · Electronic Notes in Theoretical Computer Science

An Algebra of Pure Quantum Programming

Thorsten Altenkirch, Jonathan Grattage, Juliana Kaizer Vizzotto, Amr Sabry

Published Mar 1, 2007• 37 citations
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Abstract

We develop a sound and complete equational theory for the functional quantum programming language QML. The soundness and completeness of the theory are with respect to the previously developed denotational semantics of QML. The completeness proof also gives rise to a normalisation algorithm following the normalisation-by-evaluation approach. The current work focuses on the pure fragment of QML, omitting measurements.

DOI
10.1016/j.entcs.2006.12.010

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We develop a sound and complete equational theory for the functional quantum programming language QML. The soundness and completeness of the theory are with respect to the previously developed denotational semantics of QML. The completeness proof also gives rise to a normalisation algorithm following the normalisation-by-evaluation approach. The current work focuses on the pure fragment of QML, omitting measurements.
This paper gives an introduction to and overview of the functional quantum programming language QML. The syntax of this language is defined and explained, along with a new QML definition of the quantum teleport algorithm. The categorical operational semantics of QML is also briefly introduced, in the form of annotated quantum circuits. This definition leads to a denotational semantics, given in terms of superoperators. Finally, an implementation in Haskell of the semantics for QML is presented as a compiler. The compiler takes QML programs as input, which are parsed into a Haskell datatype. The output from the compiler is either a quantum circuit (operational), an isometry (pure denotational) or a superoperator (impure denotational). Orthogonality judgements and problems with coproducts in QML are also discussed.
This paper presents a novel semantics for a quantum programming language by operator algebras, which are known to give a formulation for quantum theory that is alternative to the one by Hilbert spaces. We show that the opposite category of the category of W*-algebras and normal completely positive subunital maps is an elementary quantum flow chart category in the sense of Selinger. As a consequence, it gives a denotational semantics for Selinger's first-order functional quantum programming language QPL. The use of operator algebras allows us to accommodate infinite structures and to handle classical and quantum computations in a unified way.
The stabiliser fragment of quantum theory is a foundational building block for quantum error correction and the fault-tolerant compilation of quantum programs. In this article, we develop a sound, universal and complete denotational semantics for stabiliser operations which include measurement, classically-controlled Pauli operators, and affine classical operations, in which quantum error-correcting codes are first-class objects. The operations are interpreted as certain affine relations over finite fields. This offers a conceptually motivated and computationally-tractable alternative to the standard operator-algebraic semantics of quantum programs (whose time complexity grows exponentially as the state space increases in size). We demonstrate the power of the resulting semantics by describing a small, proof-of-concept assembly language for stabiliser programs with fully-abstract denotational semantics.
The stabiliser fragment of quantum theory is a foundational building block for quantum error correction, and hence for the fault-tolerant compilation of quantum programs. In this article, we develop a sound, universal, and complete denotational semantics for stabiliser operations, including measurement, classically controlled Pauli operators, and affine classical computation, thereby supporting an explicit treatment of quantum error-correcting codes. We interpret stabiliser operations as affine relations over finite fields, yielding a semantics that reflects the algebraic structure underlying stabiliser quantum error correction. Because stabiliser quantum mechanics has a well-behaved algebraic structure, our relational semantics is conceptually transparent and computationally tractable when compared to standard denotational models for general quantum programs. We demonstrate the resulting semantics by describing a small, low-level assembly language for stabiliser programs with fully abstract denotational semantics.
We present an extension of the denotational semantic model of the quantum programming language QML, to which computational reversibility is incorporated. The semantics of QML is defined in a functional setting which consider classical and quantum data, to which we add inverse functions. Additionally we incorporate into the semantics a history track which allows reversibility in QML. From the generation and processing of the history track and the final result of a program, the rules for executing reversibility allow to compute the original input data. This work contributes to the study of reversibility in quantum programming languages and considering that there is not yet a quantum computer in which the language can be implemented, this history and the proposed inverse functions are not trivial and allow us to determine that this language is reversible.
We develop a new framework of algebraic theories with linear parameters, and use it to analyze the equational reasoning principles of quantum computing and quantum programming languages. We use the framework as follows: we present a new elementary algebraic theory of quantum computation, built from unitary gates and measurement; we provide a completeness theorem or the elementary algebraic theory by relating it with a model from operator algebra; we extract an equational theory for a quantum programming language from the algebraic theory; we compare quantum computation with other local notions of computation by investigating variations on the algebraic theory.